Question Statement

A farmer owns a triangular shaped piece of land with corners at $X(3,7)$, $Y(6,2)$ and $Z(10,8)$. Calculate the cost of planting maize crop @ Rs. 300 per square unit.

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Background and Explanation

This problem requires finding the area of a triangle given its vertices using coordinate geometry (determinant method), then applying a per-unit rate to determine the total cost of cultivation.

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Solution

First, we find the area of the triangular field $XYZ$ using the determinant formula for the area of a triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$.

$\text{Area of triangle }XYZ=\frac{1}{2}\left|\begin{array}{ccc}3& 7& 1\\ 6& 2& 1\\ 10& 8& 1\end{array}\right|$

Expanding the determinant along the first row using cofactors:

$=\frac{1}{2}\left(3\left|\begin{array}{ll}2& 1\\ 8& 1\end{array}\right|-7\left|\begin{array}{cc}6& 1\\ 10& 1\end{array}\right|+1\left|\begin{array}{cc}6& 2\\ 10& 8\end{array}\right|\right)$

Evaluating each $2\times 2$ determinant:

$=\frac{1}{2}(3(2-8)-7(6-10)+1(48-20))$

Simplifying the expressions inside the parentheses:

$=\frac{1}{2}(-18+28+28)$

$=19\text{ sq. units}$

Thus, the cost of planting maize @ Rs. 300 per square unit is:

$\text{Cost}=19\times 300=\text{Rs. }5700$

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Key Formulas or Methods Used

• Area of triangle using determinant: $\text{Area}=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
• Cofactor expansion: Expanding a $3\times 3$ determinant along the first row
• Total cost formula: $\text{Cost}=\text{Area}\times \text{Rate per square unit}$

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Summary of Steps

1. Identify coordinates: Note the vertices $X(3,7)$, $Y(6,2)$, and $Z(10,8)$
2. Set up the determinant: Construct the $3\times 3$ matrix with $x$-coordinates in the first column, $y$-coordinates in the second, and 1s in the third
3. Expand the determinant: Apply cofactor expansion along the first row to reduce to $2\times 2$ determinants
4. Evaluate minors: Calculate each $2\times 2$ determinant (e.g., $(2-8)$, $(6-10)$, $(48-20)$)
5. Compute the area: Multiply the sum by $\frac{1}{2}$ to obtain $19$ square units
6. Calculate total cost: Multiply the area ($19$) by the rate (Rs. $300$) to get Rs. $5700$